Modeling, Simulation and Control of a 4WD

Electric Vehicle with In-Wheel Motors

R. Iervolino*, A. Sakhnevych†

*Dipartimento di Ingegneria Elettrica e delle Tecnologie dell’Informazione, Università degli

Studi di Napoli, Federico II, Via Claudio 21, 80125 Napoli, Italy, email: rafierv@unina.it

†Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli, Federico II, Via

Claudio 21, 80125 Napoli, Italy, email: aleksandr.sakhnevych@unina.it

Abstract. A relatively new technology for the electric vehicles considers the use

of brushless permanent magnet motors directly connected to the car wheels (in-

wheel motors or hub motors). In order to evaluate the performance that can be ob-

tained, a complete dynamic model of a four-wheel drive (4WD) electric vehicle

equipped with four in-wheel motors is developed and a correspondent parametric

simulator is implemented in Matlab/SimulinkTM. The simulator is also employed for

designing, testing and comparing various control logics which reproduce the han-

dling behavior of a real vehicle.

Keywords: In-wheel motors, Virtual simulation and control, Model in the Loop,

Vehicle dynamics, Differential logics.

1 Introduction

In the last years, several car manufacturers have focused their efforts on the produc-

tion of completely electric vehicles with the aim of both increasing the energy effi-

ciency and reducing the environmental impact [1, 2].

A step forward in this direction is offered by the introduction of a new technology

based on electric motors directly connected to the wheels (also known as in-wheel

motors or hub motors) [3, 4]. The best suited electric motor class for the inclusion

in the wheel rims is represented by the permanent magnet brushless DC motors,

thanks to their compactness, low weight, high torque per unit mass and volume, and

maximum speed. The adopted solutions are many, because, without the need of any

transmission shafts, traction can be moved from the front to the rear axle almost

arbitrarily. As a result, the four-wheel drive (4WD) should be easily achieved [5].

Moreover, independently controlled motors allow high dynamic performance, with-

out the presence of complex gearboxes and mechanical differentials. The price is an

increased complexity in the control design algorithms, considering also the negative

effects of augmented unsprung mass on vehicle performance dynamics [6, 7].

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In this paper, a complete vehicle, wheel and tire dynamic model is built, with the

inclusion of the electric in-wheel motors. The mathematical model is then imple-

mented in a modular and parametric simulator by using the Matlab/SimulinkTM en-

vironment. The simulator is employed both for the analysis of the system perfor-

mance and the synthesis of robust control algorithms [8, 9, 10].

The paper is organized as follows: in Section 2, some preliminaries about the

overall system mathematical model are recalled; in Section 3 the simulator is de-

scribed in detail and the most significant simulation results are reported; in Section

4 some differential control logics, related to the torque distribution among the in-

wheel motors, are analyzed and compared. Finally, Section 5 concludes the paper.

2 Preliminaries

In this Section, some preliminaries about the vehicle dynamic modeling are syn-

thetically recalled. The global mathematical model is obtained by assembling the

models of the system main components: vehicle, tires, in-wheel motors.

2.1 Vehicle dynamics

There exist numerous mathematical models in literature for representing the vehicle

dynamics. If the interest is limited to its directional behavior, with the hypothesis of

slowly varying forward speed, the bounce and the pitch motions of the car body can

be neglected [11]. By further assuming long-range and low-speed turns, the roll

degree of freedom can be also disregarded. As a consequence, a planar model is

considered for the vehicle rigid body dynamics.

Other simplifying assumptions are:

• the wheels mass is negligible;

• the vehicle is moving on a flat and rigid road, such as to be considered a geomet-

rical plane;

• there is a univocal relationship between the steering wheel angle and the wheel

angle;

• the steering axis is vertical and passes through the center of the contact.

The car body reference system has the x-axis parallel to the ground, in the forward

direction, the z-axis orthogonal to the road plan and upwards directed, and the y-

axis to complete the left-handed reference frame.

By referring to Fig. 1, with the hypothesis of small angles, the congruence equa-

tions are:

α1s = δ −v+ra1

u−rt/2 , (1)

3

α1D = δ −v+ra1

u+rt/2 , (2)

α2s = −v−ra2

u−rt/2, (3)

α2D = −v−ra2

u+rt/2, (4)

where 𝑡 is the car track width; 𝑎1 is the front wheelbase; 𝑎2 is the rear wheelbase; u

is the forwarding speed; v is the lateral speed; δ is the steering angle; 𝑟 is the yaw

rate; α1s, α1D, α2s, α2D are the slip angles.

Fig. 1 Kinematic quantities definition.

The equations of motions can be expressed by using the classical rigid body dynam-

ics equations. By referring to the force and torque scheme of Fig. 2, the model is:

m ∙ (u − vr ) = X = (Fx1s + Fx1d) ∙ cos(δ) + Fx2s + Fx2d − (Fy1s + Fy1d) ∙

sin(δ) −1

2ρCxSu2 , (5)

m ∙ (v + ur) = Y = (Fy1s + Fy1d) ∙ cos(δ) + (Fx1s + Fx1d) ∙ sin(δ) + Fy2s +

Fy2d , (6)

Jr = N = (Fy1s + Fy1d) ∙ a1 ∙ cos(δ) − (Fy2s + Fy2d) ∙ a2 + (Fx1s + Fx1d) ∙ a1 ∙

sin(δ) − 𝑡

2∙ [(𝐹𝑥1𝑠 − 𝐹𝑥1𝑑) ∙ cos(𝛿) + (𝐹𝑥2𝑠 − 𝐹𝑥2𝑑) − (𝐹𝑦1𝑠 − 𝐹𝑦1𝑑) ∙ sin(𝛿)].

(7)

4

Fig. 2 Forces acting on the vehicle.

2.2 Tire model

The interaction force between the tire and the road can be described by using the

well-known and widely employed Pacejka “Magic Formula” [12]:

Y(x) = D ∙ cos{C ∙ tan−1[B ∙ x − E ∙ (B ∙ x − tan−1(B ∙ x))]}, (8)

where Y = y + Sv , X = x + Sh, being Y the output variable, i.e. the longitudinal

force Fx or the lateral force Fy, and X the input variable, i.e. the longitudinal slip or

the lateral slip (slip angle) (see also Fig. 3).

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Fig. 3 Macro parameters of the magic formula.

In Fig. 4 the variations of Fy as a function of the the vertical load are shown.

Fig. 4 Lateral force vations for various vertical loads.

2.3 Wheel and in-wheel motor models

By assuming a rigid wheel and by neglecting the rolling resistance, the wheel equa-

tion of motions is the following:

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Ir ∙ ω = M − Fx ∙ R, (9)

where M is the applied torque, Ir is the wheel inertia, and

ω is the wheel angular velocity, Fx is the longitudinal interaction force, and R is the

wheel radius.

Fig. 5 Power and speed characteristic curves.

By neglecting the dynamics of the electric motor, a static nonlinear model is pro-

posed, based on the power and speed characteristics of the motor (e.g., see Fig. 5).

3 Differential logics

In this Section, three types of differential logics are analyzed and compared. As

a first case, the (ideal) open differential case is considered, where the engine torque

is equally distributed, independently of the real conditions of the wheels. In Fig. 6,

an example of a simulated open differential acting in presence of icy road conditions

is reported.

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Fig. 6 Slip phenomenon with in-wheel motors.

A second case considered is the speed-sensitive automatic locking differential logic.

A braking torque characteristic curve is now included in the model. Of course, the

humping phenomenon previously shown in Fig. 6 does not occur since the braking

torque suddenly goes to zero as soon as the velocity difference goes to zero.

Finally, a power sensitive automatic locking differential is analyzed. In this case it

is assumed that the braking torque is linearly increasing with the input commanded

torque. The slope of the curve is chosen considering that, as for the Torsen differ-

ential, the Torque Bias Ratio (TBR) is constant. The following relationship is

adopted:

if ωA > ωB

MA =MP

2− Mf (10)

MB =MP

2+ Mf (11)

since:

TBR =MB

MA (12)

it is:

MP

2+ Mf = TBR ∙

MP

2− TBR ∙ Mf (13)

Mf =TBR−1

1+TBR∙

MP

2 (14)

By substituting (14) in (10)-(11), it follows:

MB =TBR

1+TBR MP MA =

1

1+TBR MP (15)

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3 Simulation results

The mathematical model of the system components described above is implemented

in a Matlab/SimulinkTM simulator. In Fig. 7 the main subsystems of the simulator

are shown. Each part is characterized by a set tunable parameters and can be possi-

bly modified independently.

Fig. 7 Main subsystems of the electric vehicle simulator.

In Fig. 8, the speed sensitive and the power sensitive automatic locking differentials

are compared in a maneuver on the straight where one of the two wheels encounters

an ice floe. As it was expected, unlike the power sensitive differential, the speed

sensitive differential provides appreciable torques only when the tire begins to slip.

(a) (b)

Fig. 8 Angular velocities with a speed sensitive (a) and a power sensitive (b) differential logics.

In the following simulation results, the open and the power sensitive differential

logics are analyzed according to 4 different combinations:

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- open central and axles differentials;

- open central and automatic locking axles differentials;

- automatic locking central and open axles differentials;

- automatic locking central and axles diffentials.

Each simulation assumes a TBR = 3 for the central differential, a TBR = 5 for the

rear differential and a TBR = 2.5 for the front differential, which are typical values

for a Torsen diffential. Three different maneuvers are used to make a performance

comparison: a steering pad maneuver where the vehicle describes a circular trajec-

tory with a radius of 100 m and a velocity growing from 30km/h with a constant

acceleration of 2 m/s2; an acceleration maneuver on straight from a starting velocity

of 30 km/h and with a constant acceleration of 1.8 m/s2, in presence of ice on the

vehicle left side; an acceleration maneuver on straight from a starting velocity of 30

km/h and with a constant acceleration of 1.8 m/s2, in presence of ice on both sides

of the vehicle.

For the steering pad maneuver, the solution with both central and axles automatic

locking differentials guarantees the best curvature (see Fig. 9). Moreover, the use

of three automatic locking differentials provides the best roadholding (see Fig. 10).

Fig. 9 Steering pad maneuver.

Fig. 10 Longitudinal slip of the front left wheel during the steering pad maneuver.

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For the on straight constant acceleration maneuver, with icy conditions on the left

wheels, the cases with open differentials and with automatic locking differentials

are only reported. Again, the solution with only automatic locking differentials pro-

vides the best performance (see Figs. 11 and 12).

(a) (b)

Fig. 11 Front (a) and rear (b) wheels angular velocities for an on straight acceleration maneuver

with icy conditions on the left side of the vehicle and open differentials .

(a)

(b)

Fig. 12 Front (a) and rear (b) wheels angular velocities for an on straight acceleration maneuver

with icy conditions on the left side of the vehicle and automatic locking differentials.

Similar results are obtained in last simulation scenario, i.e. an on straight constant

acceleration maneuver with icy conditions on both sides of the vehicle. It is con-

firmed that the solution with central and axles automatic locking differentials guar-

antees the best performance.

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4 Conclusions

In this paper the influence of the in-wheel motors on the vehicle dynamics has been

analyzed. Starting from the mathematical model of the system components, a mod-

ular and parametric simulator has been developed in Matlab/SimulinkTM. The cru-

cial problem of the torque distribution control among the 4 electric motors has been

considered by evaluating various differential logics which mimic the currently em-

ployed ones. From the simulation results for the considered maneuvers, the best

solution consists in adopting an automatic locking logic for both central and axles

diffentials, which can be more easily implemented with respect to the internal com-

bustion engines case. Possible future developments of this work could be the inclu-

sion of an ESP control system with also a more complex car model for both the

sprung and the unsprung masses and moreover the study of torque distribution

logics not deriving from existing mechanical devices.

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